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Theorem 0ss 3401
Description: The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
0ss  |-  (/)  C_  A

Proof of Theorem 0ss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 noel 3367 . . 3  |-  -.  x  e.  (/)
21pm2.21i 635 . 2  |-  ( x  e.  (/)  ->  x  e.  A )
32ssriv 3101 1  |-  (/)  C_  A
Colors of variables: wff set class
Syntax hints:    e. wcel 1480    C_ wss 3071   (/)c0 3363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364
This theorem is referenced by:  ss0b  3402  ssdifeq0  3445  sssnr  3680  ssprr  3683  uni0  3763  int0el  3801  0disj  3926  disjx0  3928  tr0  4037  0elpw  4088  exmidsssn  4125  fr0  4273  elnn  4519  rel0  4664  0ima  4899  fun0  5181  f0  5313  oaword1  6367  0domg  6731  nnnninf  7023  exmidfodomrlemim  7057  sum0  11164  ennnfonelemj0  11920  ennnfonelemkh  11931  0opn  12182  baspartn  12226  0cld  12290  ntr0  12312  bdeq0  13118  bj-omtrans  13207  el2oss1o  13241  nninfsellemsuc  13261
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